TMDPDF: A proper DefinitionAnd Its Evolution

Ahmad Idilbi University Of Regensburg QCD Evolution 2012 Workshop Jefferson LAB, May 15

M. G. Echevarria, A. Idilbi, I. Scimemi, arXive: 1111.4996[hep-ph]

OutlineRelevance Of TMDPDFs/TMDFFs For High-Energy

ProcessesProblems With Existing Definitions``New Definition’’: On-The-LC vs Off-The-LC Of

CollinsProperties: Free From Light-Cone Singularities,

Universality, Gauge Invariance, PDF From TMDPDF

Evolution: Anomalous Dimension At Second And Third Orders In The Strong Coupling

Conclusions And Outlook

Factorization For q_T-Dependent Observables

Collinear Matrix Element:

Soft Function:

q_T Dependent Observables/MESemi-Inclusive DIS: Factorize Into TMDPDF, Fragmentation Function and Soft

Function With All Quantities Have TMD [Ji, Ma, Yuan,03]

q_T-Higgs Boson Production [With Or Without Jets Production] [Gluon TMDPDF]

Plenty (Or A ``Zoo’’) Of Such ME. However The TMDPDF Although Is The Simplest. Its ``Accepted’’ Definition Remains Illusive For Almost 30yr

Collins-Soper TMDPDF [Collins-Sopers,82]Partonically:

Problems: Light-Cone Divergences

• Real Gluon Emission Has The Same Pathology But No Rescue!• For The PDF We Have Similar Light-Cone-Singularities But ``Things Are Fine’’. It Is easy To Understand Why!

THE ``TMDPDF’’ Is Not Well-Defined Even After Regulation!!

The Vertex With The Wilson Line:

Soft Function: Similar Issues

THE Soft Function Is Also Not Well-Defined

The Issue Of Double-Counting•The Hadronic Tensor , M, Calculated (Partonically) In Full QCD Is Free From Mixed Divergences•The Supposedly ``Collinear’’ MEs Are Plagued With Soft Contamination!

Definition Of The TMDPDF• To Cancel The Mixed Divergences , Avoid Double Counting Among Soft And Collinear And As Imposed By SCET Power Counting We Define The TMDPDF

• Six Different Definitions of TMDPDF1. Collins 82, Collins 20112. Ji, Ma and Yuan 033. Cherednikov and Stefanis 08- 4. Mantry and Petriello, 20105. Becher and Neubert, 20106. Rothstein et.al 2011

TMDPDF ON-The-Light-Cone (I)

TMDPDF On-The-Light-Cone (II)

The Real Gluon Contribution Is Free From Mixed Divergences

Anomalous Dimension Of TMDPDF

[Manohar 03,Idilbi, Ji 03]

• The TMDPDF—On The LC—Evolves ONLY With The Hard Probe.• No Extra Auxiliary (And Non-Physical Parameters) Are Needed!• Going Beyond One-Loop Order Needs More Sophistication (No Brute Force Is Needed!)

Universality(I)

Universality Of The Soft Function

Universality Of The Collinear Function

Gauge InvrianceHadronic ME Should Be Gauge Invariant!This Issue Is Especially Tricky when TMD Is

Present!Consider The Light-Cone Gauge

Ji et.al 03, CS 08--,M.E, A.I, I.S,10

The Collinear And Soft Function Are Identical –At The Integrand Level In Mandelstam-Leibbrandt Prescription. Thus Gauge Invariance Is Realized

From TMDPDF To PDF

Operator Product Expansion And The Q^2 Dependence

For Perturbative q_T

At Higher Orders In PT The Q^2-Dependence Cannot Be eliminated By Any Single Choice Of The Intermediate Scale. Thus Their Resummation Is A Must

Q^2-ResummationFrom The Definition Of The TMDPDF

By Dimensional Arguments And Lorentz Invariance

Recall:

Since The PDF And The Wilson Coefficient Are Free From Rapidity DivergencesTo All Orders In PT

Q^2-Resummation Continued

The Naïve Collinear And The Soft Functions Are Linear In Their Last Arguments

Q^2-Independent

All The Large Logarithms of Q^2 Are Resummed In First Exponent

The D-Term(I)

[A Is Twice The Cusp Anomalous Dimension]

Recursion Relations Which Uniquely Determine The D-Term

The D-Term(II)By Knowing The Cusp Anomalous Dimension At Three Loop Order

AD At Higher Orders In PT•The AD Of H Is Known From Previous Full QCD Calculations Up to Third Order In The strong Coupling. Based On The Factorization Theorem And RG Invariance:

[Idilbi, Ji and Yuan 06]

The A And B Are Known To Third Order! So We Can Extract The AD Of The TMDPDF To Third Order

Putting Things TogetherResummed TMDPDF

Evolved And Resummed TMDPDF

All Large Logarithms Of Q/q_T Are Resummed In The Exponential Factors!

Collins Approach

[Collins 2011]The Collinear Part Is Kept On-The-LC While Only The Soft FunctionIs ``Tilted’’. With This

Collins TMDPDF Evolves With Two Variables. One Is Un-Physical!

Compare With:

If You Stay On-The-LC You Only Run With Hard Probe Q!

Going Off-The-LC Is Meant To Regularize Rapidity Divergences. This Can Be Achieved While Staying On The LC.

There Is No Natural Choice Of y_n That Takes You Back To The LC!!

ConclusionsWe Introduced A TMDPDF Defined On-The-

Light-Cone Emphasizing The Role Of The Soft Function Contribution

General Features: Free From RD, Enables Us To Recover The Integrated PDF, Gauge Invariance And Universality

The Evolution Is Controlled Only By The Hard Probe

Higher Order ADs Are Easy To Extract Based On Factorization Theorems.

Going Of-The-LC Has No Special ImportanceGeneralizations To Other TMD Quantities Is

Straightfoward